interaction of two swimming paramecia · paramecium caudatumwas used in this study, because the...

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IntroductionIt is important to understand and predict the flow of

suspensions of micro-organisms that appear in massiveplankton blooms in the ocean, harmful red tides in coastalregions and are used in bioreactors. The size of individualmicro-organisms is often much smaller than that of the flowfield of interest, so the suspension is modelled as a continuumin which the variables are volume-averaged quantities (Fashamet al., 1990; Pedley and Kessler, 1992; Metcalfe et al., 2004).Continuum models for suspensions of swimming micro-organisms have also been proposed for the analysis ofphenomena such as bioconvection (e.g. Childress et al., 1975;Pedley and Kessler, 1990; Hillesdon et al., 1995; Bees and Hill,1998; Metcalfe and Pedley, 2001). However, the continuummodels proposed so far are restricted to dilute suspensions, inwhich cell–cell interactions are negligible. If one wishes toconsider larger cell concentrations, it will be necessary toconsider the interactions between micro-organisms. Then, thetranslational–rotational velocity of the micro-organisms, theparticle stress tensor and the diffusion tensor in the continuummodel will need to be refined.

To understand the interactions between micro-organisms, itis first necessary to clarify two-cell interactions. Thus, weinvestigated the interaction between two model micro-organisms analytically (Ishikawa et al., 2006), assuming thatthe cell–cell interaction is purely hydrodynamic; no biologicalreactions were considered. In practice, however, it is to beexpected that in the presence of a nearby micro-organism, agiven micro-organism would not behave as if it were alone. A

micro-organism may consider reproducing sexually orattempting to consume (or avoid being consumed by) itsneighbour. It may also move away from it, because of theincreased competition for food. Although two-cell interactionsare important when considering the interactions between manycells, it is not at all clear how cells behave when they are inclose contact. Also, cell–cell interactions were not modelledprecisely in any previous analytical studies dealing with a non-dilute suspension of micro-organisms (Guell et al., 1988;Ramia et al., 1993; Nasseri and Phan-Thien, 1997; Lega andPassot, 2003; Jiang et al., 2002). In this study, we clarify howtwo micro-organisms interact when they come close to eachother, both experimentally and numerically.

Paramecium caudatum was used in this study, because thebehavior of individual cells of this organism is well understood.Naitoh and Sugino investigated two types of biologicalreactions of a solitary Paramecium cell to mechanicalstimulations (Naitoh and Sugino, 1984). (i) Avoiding reactionsoccur when a cell bumps against a solid object with its anteriorend. The cell swims backward first, gyrates about its posteriorend, and then resumes normal forward locomotion. (ii) Escapereactions occur when the cell’s posterior end is mechanicallyagitated. The cell increases its forward swimming velocity fora moment, then resumes normal forward locomotion. Thechange in the swimming motion is regulated by changes inmembrane potential, because Paramecium cells, like othermonads, have no nerves for transmitting stimulativeinformation nor synapses to determine transmission direction.Machemer clarified the frequency and directional responses of

The interaction between two swimming Parameciumcaudatum was investigated experimentally. Cell motionwas restricted between flat plates, and avoiding and escapereactions were observed, as well as hydrodynamicinteractions. The results showed that changes in directionbetween two swimming cells were induced mainly byhydrodynamic forces and that the biological reaction was aminor factor. Numerical simulations were also performedusing a boundary element method. P. caudatum was

modelled as a rigid spheroid with surface tangentialvelocity measured by a particle image velocimetry (PIV)technique. Hydrodynamic interactions observed in theexperiment agreed well with the numerical simulations, sowe can conclude that the present cell model is appropriatefor describing the motion of P. caudatum.

Key words: hydrodynamic interaction, Paramecium caudatum,biological reaction, swimming motion, numerical simulation.

Summary

The Journal of Experimental Biology 209, 4452-4463Published by The Company of Biologists 2006doi:10.1242/jeb.02537

Interaction of two swimming Paramecia

Takuji Ishikawa1,* and Masateru Hota2

1Department of Bioengineering and Robotics, Graduate School of Engineering, Tohoku University, Aoba 6-6-01,Sendai 980-8579, Japan and 2Department of Mechanical Engineering, University of Fukui, 3-9-1 Bunkyo,

Fukui 610-8507, Japan*Author for correspondence (e-mail: [email protected])

Accepted 8 September 2006

THE JOURNAL OF EXPERIMENTAL BIOLOGY

4453Interaction of two swimming Paramecia

cilia to changes in membrane potential in Paramecium cells(Machemer, 1974). There are many Ca2+ channels in theanterior end, whereas there are many K+ channels in theposterior end. The locality of the ion channels is the essentialmechanism controlling Paramecium’s biological reaction(Naitoh and Sugino, 1984). In reality, the range of micro-organism lengths is large, and they alter their behaviouraccording to many environmental parameters. The variety ofshapes both within and between species is also vast (Brennenand Winet, 1977). Many types of ciliate, however, show asimilar biological reaction to that of Paramecium cells, becausethey also have Ca2+ channels in the anterior end and K+

channels in the posterior end. We focused on Paramecium cellsin the present study, but we expect that our results will beapplicable to other ciliates and micro-organisms.

The interaction between two P. caudatum cells wasinvestigated both experimentally and numerically. Cell motionwas restricted between flat plates with a gap of about 70·mm.We observed avoiding and escape reactions as well ashydrodynamic interactions. To conclude that the interaction ispurely hydrodynamic, it is necessary to simulate the cells’motion numerically without any biological response. Thus, theP. caudatum cell was modelled as a rigid spheroid withprescribed surface tangential velocities measured by a particleimage velocimetry (PIV) technique. The model is referred toas a ‘squirmer’, and is explained briefly in Materials andmethods (for details, see Ishikawa et al., 2006). The interactionbetween two squirmers was calculated by a boundary elementmethod, and the results were compared with those of theexperiments. The authors have used a squirmer model for otherstudies (Ishikawa et al., 2006; Pedley and Ishikawa, 2004)(J. Ishikawa and T. J. Pedley, manuscript submitted forpublication), so a comparison was also made with the earlierdata to check the reliability of the squirmer model.

Materials and methodsMaterials and experimental setup

Cultures of the unicellular freshwater ciliate Parameciumcaudatum Ehrenberg were used for the experiments. They weregrown by inoculating 2·ml of a logarithmic-phase culture into250·ml of culture fluid. The culture fluid was made by boilingdemineralised water with 2·g of straw, then adding 0.2·g ofyeast powder. It was kept at 20°C for 2 weeks beforeinoculation. Measurements were performed 2 weeks afterinoculation of the culture at 19–21°C. Microscopic observationshowed that the body length of an individual cell was in therange of approximately 200–250·mm and the widthapproximately 40–50·mm. The swimming speed of anindividual cell was approximately 1·mm·s–1. The swimmingmotion of P. caudatum in a free space is not straight, but formsa left spiral. The pitch of the spiral was approximately 2·mmand the width was 0.4·mm.

The experimental setup was designed to measure thedisplacements of cells in a still fluid between flat plates (Fig.·1).The motion of cells was restricted to two-dimensions for the

following two reasons: (i) the orientation vector of the cell andthe distance between two surfaces of cells were easy to measurewithout a considerable error, and (ii) biological reactions wereobserved much more frequently in two-dimensional space,because cells experienced strong collisions when there was noheight variation. The latter reason is important in order toanalyse a large number of biological reactions. Theexperimental setup consisted of a digital video (DV) camerawith a 243 macrolens, light sources, and inner and outerdishes. The test fluid was placed between the top of the outerdish and the bottom of the inner dish. The gap between the twodishes was about 70·mm, so that cells could not overlap three-dimensionally. The test fluid was same as the culture fluid, butthe volume fraction of cells was adjusted to within about0.5–1% so that three-cell interactions rarely occurred. Cellmovements were recorded by the DV camera over a 10–20·minperiod, and we did not observe any decrease in the cells’swimming velocity or aggregation due to the cell chemotaxisduring the measurements.

Sample sequences of the interaction between two swimmingP. caudatum observed in the experiment are shown in Fig.·2.The time interval for each sequence is Dt=1/3s, and thebackground is subtracted from the figure by image processing.We observed such interactions several times per 10–20·minexperiment. Since the intention was to concentrate on two-cellinteractions, data were deleted if there was a third cell within750·mm of one of the two interacting cells. The velocitydisturbance caused by a force-free cell near a wall boundarydecays as r–2, where r is the distance from the cell, so the effectof the third cell on the swimming velocities of the twointeracting cells is less than about 10% if it is farther than750·mm away. (The derivation of velocity disturbance due to aforce-free particle near a wall boundary is shown in AppendixA.) Microscopic observation showed that some solitary cellsoccasionally stopped swimming, even though there seemed tobe no mechanical stimulation. They then swam backward,gyrated about their posterior ends, and finally resumed normalforward locomotion. Such a reaction is rare if one uses a culture

Digital video camera

Light source Light source

Inner dishOuter dish

Test fluid

Macro lens

70 μm

Fig.·1. Schematics of the experimental apparatus. Test fluid wasplaced between the bottom of the inner dish and the top of the outerdish.


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2 weeks after the inoculation, but could not be eliminatedcompletely. Since we intended to measure two-cell interactions,data were deleted if one of two interacting cells showed this typeof solitary reaction before the two cells approached within adistance of L, where L is the body length of the shorter cell. Theinteraction data was recorded regardless of whether it was abiological reaction or a hydrodynamic interaction, and the totalnumber of data recorded in this study was 301.

Surface velocity of P. caudatum

The velocity field around a swimming P. caudatum cellbetween flat plates was measured using a PIV technique. Asmall amount of milk was added to the culture fluid as tracerparticles, and the flow field was recorded using a CMOScamera (Basler A602f; 6593493·pixels, 100·frames·s–1; BaslerAG, Ahrensburg, Germany) through a microscope. A sampleoriginal image is shown in Fig.·3. The velocity vector wascalculated from two successive images using a MatPIV (Grueet al., 1999), in which spurious vectors were removed by asignal-to-noise ratio filter (cf. Keane and Adrian, 1992), aglobal histogram operator and a median filter. The spuriousvectors were replaced by vectors linearly interpolated fromneighbouring points. The velocity vector was calculated for atime series of images, and approximately 80·000 velocityvectors around a swimming cell were obtained. These vectorswere averaged by assuming that the velocity field isaxisymmetric and time-independent. The velocity field relativeto the cell’s swimming velocity vector is shown in Fig.·4. Wesee that the cell swims smoothly without generating arecirculation region.

The surface velocity of P. caudatum has to be measured foruse as a boundary condition for the numerical simulation, asexplained in below. We intended to measure the surfacevelocity of the cell by the PIV technique; however, the methodshowed considerable instability at short wavelengths, so wecould not accurately measure the velocity vector using thismethod. Thus, we interpolated surface velocity as follows. A

T. Ishikawa and M. Hota

cell was assumed to be an extended ellipsoid with a minor axisof 0.36, where length was nondimensionalised relative to themajor axis. The surface velocity vectors were interpolatedlinearly from those at 0.18 and 0.36 from the surface with thesame angle u from the orientation vector of the cell. Weassumed that surface velocity was mainly tangential, because

Fig.·2. Sequences showing the interaction of two swimming P.caudatum cells observed in the experiment. The background wassubtracted from the figure. Fig.·3. Original image for a swimming P. caudatum cell in a water

with a small amount of milk between flat plates.

Fig.·4. Velocity vectors relative to the swimming velocity of P.caudatum, which were calculated by the PIV methods by assumingthat the velocity field is axisymmetric and time-independent. The largearrow indicates the swimming direction of the cell.



cells did not deform and fluid did not penetrate its surface athigh velocities. The tangential surface velocity, us, interpolatedfrom the PIV data is shown in Fig.·5. We see that us is fasternear the anterior end than near the posterior end.

The experimental results of us were approximated by:

where coefficients ci were determined by the method ofleast squares (c1=1.707, c2=0.2400, c3=0.2472, c4=0.1506,c5=0.1154). us (Eqn·1) is also shown in Fig.·5. The high modesdecay faster than the low modes as the distance from the cellincreases; thus, the far-field velocity is governed by the lowestmode (cf. Ishikawa et al., 2006). Moreover, the role of highmodes in the near-field is to generate fluctuations in velocity.Hence, the overall properties, such as the trajectories of a pairof cells, may be captured by the first few modes. Thus, we usedup to the fifth mode in Eqn·1, which was used as a boundarycondition for the cell surface (cf. Numerical methods, below).

i=1

ci sin(i�) ,us � 0 � � � � ,

5

(1)

Numerical methods

The P. caudatum cell was modelled as a rigid spheroid withthe surface tangential velocity measured by the PIV technique,referred to as a ‘squirmer’. The squirmer model was firstproposed by Lighthill (Lighthill, 1952), and his analysis wasthen extended (Blake, 1971a). The numerical methods used inthis study are similar to those reported elsewhere (Ishikawa etal., 2006), so only a brief explanation is given here. A squirmeris assumed to be neutrally buoyant and torque-free. TheReynolds number based on the swimming speed and the radiusof individuals is small, so that the flow field can be assumed tobe a Stokes flow. Brownian motion is not taken into account,because a cell is too large for Brownian effects to be important.The spheroid’s surface is assumed to move purely tangentiallyand this tangential motion is assumed to be axisymmetric andtime-independent. The tangential surface velocity of a squirmeris given by Eqn·1. We performed a trial simulation for a solitarysquirmer swimming in a still fluid with the boundary conditionof Eqn·1. The results showed that the swimming velocity of thesquirmer was 1.04, although it should be 1.0 because the

1 2 3

0.5

1

1.5

0

us

Experimental resultsApproximated function

�

Fig.·5. Experimental results and an approximated curve defined byEqn·1 for a surface velocity of P. caudatum. The coefficients in Eqn·1are c1=1.707, c2=0.2400, c3=0.2472, c4=0.1506 and c5=0.1154.

Fig.·6. Computational mesh for two interacting squirmers, in which590 triangle elements are generated per squirmer. The mesh is finerin the near-contact region. Using the boundary element method, thecomputational mesh is generated only on the particle surfaces.

1

2

Escape reaction (Δt=1/3 s)

1

2

Avoiding reaction (Δt=1/6 s)

B

A

Fig.·7. Sequences showing the biological reactions when twoswimming P. caudatum (labeled 1, 2) experience a near-contact. Longarrows are added to schematically show cell motion. (A) Avoidingreaction (Dt=1/6·s). (B) Escape reaction (Dt=1/3·s). Scale bars,500·mm.


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surface velocity is non-dimensionalized by the swimmingvelocity. The error may arise from the us approximation,although it is very small.

When there are two squirmers in an infinite fluid otherwiseat rest, the Stokes flow field external to the squirmers can begiven in an integral form as:

where Am is the surface of squirmer m and K is the Oseentensor. The single-layer potential, q, is the subtraction of thetraction force on the inner surface from that on the outersurface. The boundary condition is given by:

u(x) = Um + Vm ` (x–xm) + us,m , xP Am , (3)

where Um and Vm are the translational and rotational velocitiesof squirmer m, respectively. xm is the centre of squirmer m, andus,m is the squirming velocity of squirmer m, given by Eqn·1.In simulating hydrodynamic interactions between twosquirmers, we assume that the surface velocity is independentof the distance between the cells. Thus, no biological reactionis modelled, and the interaction is purely hydrodynamic.Similarly to Ishikawa et al. (Ishikawa et al., 2006), we

Am

⌠⎮⌡

m=1

K(x–x) . q(x)dAm ,u(x) = –

2

(2)


employed another boundary condition of constant swimmingpower; however, the difference in trajectories was very small(see Appendix B).

In the experiment, two cells swim between flat plates, andthe motion of cells was restricted to a two-dimensional plane.In the numerical simulation, on the other hand, squirmers swimin an infinite fluid, but the centres and orientation vectors oftwo squirmers were initially set in the same plane. The motionof squirmers remained in the plane, even though the flow fieldis three-dimensional, because the surface squirming velocityis axisymmetric. The two flat plates were omitted in thesimulation, because adding them does not qualitatively affectthe interaction provided that the two squirmers remain in thesame plane. The quantitative effect of wall boundaries on theswimming speed was also small (see Appendix C).

The boundary element method was employed to discretizeEqn·2. The computational mesh used in this study is shown inFig.·6. A maximum of 590 triangle elements per particle weregenerated, and the mesh was finer in the near-contact region.Time-marching was performed by 4th-order Runge–Kuttaschemes. A non-hydrodynamic interparticle repulsive force,Frep, was added to the system in order to avoid the prohibitivelysmall time step needed to overcome the problem of overlappingparticles. We followed published methods (Brady and Bossis,

1

2

1

2

1

2

1

2

Sample case 3Sample case 1

B

A

D

C

Sample case 2 Sample case 4

Fig.·8. Some sample sequences showing the hydrodynamical interactions when two swimming P. caudatum experience a near-contact. The timeinterval between each sequence is 1/3·s. Long arrows are added to schematically show cell motion. (A) Sample case 1; (B) sample case 2; (C)sample case 3; (D) sample case 4. Scale bars, 500·mm.



1985) (T. Ishikawa and T. J. Pedley, manuscript submitted forpublication), and used the following function:

where a1 is a dimensional coefficient, and a2 is a dimensionlesscoefficient. e is the minimum separation between two surfaces,which was calculated by the iterative methods proposed(Claeys and Brady, 1993). The separation vector h connects theminimum separation points on the two surfaces. Thecoefficients used in this study were a1=0.1 and a2=103. Theminimum separation obtained using these parameters was in

�2 exp(–�2�)Frep = �1

1 – exp(–�2�)

h

h, (4)

the range 10–3–10–4. The effect of the repulsive force on thetrajectories of cells was very small, because it acts only in thevery near field and changes the distance between particles byonly approximately 10–4. This was confirmed numerically bycomparing three trajectories, which are shown in Figs·9–11,with and without a repulsive force.

ResultsExperimental results

Before analysing the results of two-cell interactions, weshow some typical observation results in order to help readersunderstand the phenomena. Fig.·7A shows an avoidingreaction, in which arrows are drawn to schematically show cellmotion. The time interval for each sequence is Dt=1/6s and thecolour is reversed from the original figure (as shown in Fig.·2).We see that two cells collide with their anterior ends. They firstswim backward, then gyrate about their posterior ends, thenresume normal forward locomotion. We defined an avoidingreaction as one involving backward swimming during theinteraction.

Fig.·7B shows an escape reaction, which occurs when thecell’s posterior end is strongly agitated. We see that cell 2increases its forward swimming velocity after the collision. The

original movie was taken 30·frames·s–1, andthe velocity vector of a cell was calculated bytracking the centre of the major axis of the cellin each frame. If a cell’s swimming velocityafter the collision became 1.3 times fasterthan the initial value, we classified theinteraction as an escape reaction, where theinitial value was defined when two cells’surfaces were at a distance of L before thecollision.

When a cell–cell interaction did notsatisfy the definition of an avoiding reaction(AR) nor that of an escape reaction (ER), itwas classified as a hydrodynamic interaction(HI). Four kinds of typical HI results areshown in Fig.·8. When two cells are initiallyfacing, they come close to each other at first,then they change their directions slightly inthe near field, and finally move away from

Table·1. Ratio of three types of interaction

Number Percentage Type of interaction of cells (%)

Hydrodynamic interaction (HI) 510 84.7Avoiding reaction (AR) 29 4.8Escape reaction (ER) 63 10.5

Total number of experimental cases recorded in this study, 301;total number of cells, 602.

Fig.·9. Sequences (A–F) showing thehydrodynamic interactions between two squirmersunder the initial condition of uin<p. At t=0, thereis a distance of 0.3 in the perpendicular directionto the orientation vectors, where t is thedimensionless time and t=0 is the initial instant.The orientation vectors of the squirmers are shownas large arrows on the ellipsoids, and a thin solidline is added so that one can easily compare theangle between the two squirmers. du, explained inFig.·12, is about 0.0 in this case. (A) t=0.0; (B)t=1.0; (C) t=1.5; (D) t=2.0; (E) t=3.0; (F) t=5.0.


4458

each other (see Fig.·8A). The change in direction is small inthis case. When the initial angle between the orientationvectors of the two cells is less than about 3p/4, there are twomain kinds of swimming motions. If one cell collides with theposterior end of the other cell, the two cells do notsignificantly change their swimming direction, as shown inFig.·8B. On the other hand, if one cell collides with theanterior end of the other cell, the two cells tend to swim sideby side at first, then move away from each other with an acuteangle, as shown in Fig.·8C,D. The final angle in this caseseems to be independent of the initial angle, which will bediscussed in detail below.

All the interacting cells were classified as HI, AR or ER, andthe results are shown in Table·1. The total number ofinteracting cells recorded in this study was 602 (the totalnumber of cases was 301). It was found that 84.7% of cellsinteract hydrodynamically. The ratio of ER is slightly higherthan that of AR.

Numerical results

Although we have assumed that the experimental data notsatisfying the definition of AR nor ER are HI, there is noevidence that two cells do not actively change their swimming


motions during the interaction. Thus, we also performednumerical simulations. The authors used a squirmer model forsome previous studies (Ishikawa et al., 2006; Pedley andIshikawa, 2004) (T. Ishikawa and T. J. Pedley, manuscriptsubmitted for publication), so a comparison with the earlier datawas also made to check the reliability of the squirmer model.

We first show the interactions between two squirmers withuin<0 with a small distance between the squirmers in theperpendicular direction to the orientation vectors, as shown inFig.·9. The orientation vectors of the squirmers are shown aslarge arrows on the ellipsoids, and a thin solid line is added sothat one can easily compare the angle between the two squirmers.It is found from the figure that the two squirmers come very closeto each other, then change their orientation in the near field, andfinally move away from each other. This tendency is similar tothe experimental results shown in Fig.·8A.

Fig.·10 shows the interactions between two squirmers withuin<2p/3, in which one squirmer collides with the posterior endof the other. We see that the two cells do not significantly changetheir swimming direction. This tendency is the same as theexperimental results shown in Fig.·8B. If one squirmer collideswith the anterior end of the other, on the other hand, the two cellstend to swim side by side at first, then move away from each

other with an acute angle, as shown in Fig.·11.This tendency is again the same as theexperimental results shown in Fig.·8C,D.

DiscussionWe can conclude from Table·1 that the

cell–cell interaction is mainly hydrodynamicand that the biological reaction accounts for aminority of incidents. In the presentexperiment, the motion of cells was restrictedbetween flat plates. In a real cell suspension,however, cells often swim freely in three-dimensional space. Since the effect of thewalls on the cell behaviour is unclear, weperformed a trial experiment in which cellsswim three-dimensionally in a largecontainer. In the trial experiment, weobserved fewer biological reactions, becausecells could avoid each other using threedimensions, and strong collisions occurredinfrequently. We expect, therefore, that the

Fig.·10. Sequences (A–F) showing thehydrodynamic interactions between twosquirmers under the initial condition of uin<2p/3.The orientation vectors of the squirmers areshown as large arrows on the ellipsoids, and a thinsolid line is added so that one can easily comparethe angle between the two squirmers. du,explained in Fig.·12, is about 0.3 in this case. (A)t=2.0; (B) t=3.2; (C) t=4.0; (D) t=4.4; (E) t=5.0;(F) t=6.0.



Fig.·11. Sequences (A–F) showing thehydrodynamic interactions between two squirmersunder the initial condition of uin<p/2. Theorientation vectors of the squirmers are shown aslarge arrows on the ellipsoids, and a thin solid lineis added so that one can easily compare the anglebetween the two squirmers. du, explained inFig.·12, is about 2.0 in this case. (A) t=0.0; (B)t=2.4; (C) t=3.0; (D) t=4.5; (E) t=6.0; (F) t=9.0.

dθ

e1,out1

2

L

L

L

θout

e2,out

e1,in

e2,in

e2,oute1

e2,in

θin

Fig.·12. Definition of three kinds of angles, where e1 and e2 are theorientation vectors of cell 1 and 2, respectively. uin is the angle betweene1,in and e2,in when two cells surfaces are at a distance of L before thecollision. uout is the angle between e1,out and e2,out when two cell surfacesare at a distance of L after the collision. In order to describes the changeof orientation of cell 2 relative to cell 1, a frame is fixed to cell 1 so thatthe frame rotates when cell 1 rotates, and du is defined as change in theangle of cell 2 relative to this rotating frame.

0

10

20

30

40

θin

Rea

ctio

n ra

tio (

%)

0 π/2 π

ARHead–headHead–bodyHead–tail

ER Head–tail

Fig.·13. Change of the reaction rate with uin. AR, avoidingreaction; ER, escape reaction. The data are classified according tothe contact points. Head–tail, for instance, indicates that thecollision occurs between the head of cell 1 and the tail of cell 2(cf. Fig.·14).


4460

conclusion obtained here is also valid for cell–cell interactionsin a cell suspension.

In order to analyse the experimental results quantitatively,we defined three angles during cell–cell interactions (seeFig.·12). uin is the angle between e1,in and e2,in, where ei,in is theorientation vector of cell i when the two cell surfaces are at adistance of L before the collision. uout is the angle between e1,out

and e2,out, where ei,out is the orientation vector of cell i whenthe two cell surfaces are at a distance of L after the collision.In order to describe the change in orientation of cell 2 relativeto cell 1, a frame is fixed to cell 1 so that the frame rotates whencell 1 rotates, and du is defined as the change in the angle ofcell 2 relative to this rotating frame. Data for AR and ER areclassified into seven uin ranges as well as three contactpositions, where a cell is divided into three equal-lengthsections; head, body and tail, respectively from the anteriorend. The reaction rates for each uin range and each contactposition are shown in Fig.·13, where reaction rates arecalculated by dividing the number of AR or ER for a certainuin range and a certain contact position by the total number of


cells under the same conditions. In Fig.·13, head–tail, forinstance, indicates that the collision occurs between the headof one cell and the tail of the other (see Fig.·14). We see thatAR occurs mostly when uin2p/3 and under head–head orhead–body conditions. AR rarely occurs under a head–tailcondition. This is apparently because there are many Ca2+

channels in the anterior end, as explained in the Introduction.ER occurs only when one cell collides with the tail of the

other, and the reaction rate is not sensitive to uin, as shown inFig.·13. The dependence of contact position can be explained bythe locality of K+ channels, as explained in the Introduction. Thereaction rate of ER is higher than that of AR, so we can say thatER occurs more frequently in the cell–cell interaction. Thetemporal change of the dimension-free swimming velocity under

Head

Head

Body

Body

TailTail

0 0.2 0.4 0.6 0.8 1

1

2

3

t (s)

U/U

in

Fig.·15. Temporal change of swimming velocity in the case of escapereaction. Error bars show the standard deviation of 63 escaping cells.Collision occurs at t=0, and Uin is the velocity when two cell surfacesare at a distance of L before the collision.

Fig.·14. Head–tail interaction, in which the collision occurs betweenthe head of one cell and the tail of the other. A cell is divided intothree equal length sections; head, body and tail, respectively, from theanterior end.

dθ

π/2

–π/2

0

πy=x

θin

0 π/2 π

Head–headHead–bodyHead–tail

Fig.·16. Correlation between uin and du for the three contact positions.The broken line of slope one and the solid line of du=0 are added forcomparison.

Experiment Simulation

y=x+0.4

dθ

π/2

–π/2

0

π

y=x

θin

0 π/2 π

Head–headHead–bodyHead–tail

Head–bodyHead–tail

Fig.·17. Comparison of the results of du between the experiments andthe simulations. Gray symbols, experimental results; the numericalresults are plotted by large circles and squares. The broken line ofslope one and the solid lines of du=0 and du=uin+0.4 are added forcomparison.



ER was measured, and the results are shown in Fig.·15. Sincethe escaping velocity depends on how strongly the posterior endof the cell is agitated, the error bar becomes very long. We seefrom the figure that the escaping velocity is maximum att=0.2–0.3·s, and its value is about 1.8 times the initial velocity.

When a cell–cell interaction does not satisfy the definition ofAR or ER, it is classified as HI. The correlations between du anduin for HI for the three contact positions are shown in Fig.·16.We see that du for head–tail (white squares) is distributed neardu=0, which means that the two cells do not change theirorientations significantly after collision (cf. Fig.·8B). The casesfor head–head and head–body, however, show a differenttendency. When uin is smaller than approximately 3p/4, duincreases almost linearly with uin. A broken line with slope oneis drawn in Fig.·16 for comparison. We see that du is slightlylarger than uin, which means that the two cells tend to swim sideby side at first, then move away from each other with a smallangle, as shown in Fig.·8C,D. When uin is larger thanapproximately 3p/4, du is distributed near du=0, which againmeans that the two cells do not change their orientationssignificantly (cf. Fig.·8A). Consequently a tendency of thehydrodynamic interaction between two cells clearly appears, andis dependent on uin and contact position.

In order to compare numerical and experimental resultsquantitatively, we performed simulations for various uin

conditions and for two contact positions (head–head andhead–tail). The correlations between du and uin obtained in thesimulations are shown in Fig.·17. We see that du for head–tail(large white squares) is distributed near du=0, which is thesame tendency as in the experiments. In the case of head–head(large white circles), du increases almost linearly with uin whenuin is smaller than approximately 3p/4. This tendency is alsothe same as in the experiments. The final angle uout for thesesquirmers is about 0.4·rad; thus, the numerical results fit wellwith y=x+0.4, as shown in the figure. When uin is larger thanapproximately 3p/4, du is distributed near du=0, which is againthe same as in the experiments. We can conclude, therefore,that the HI data in the experiments agree well with thenumerical results and that the interaction is purelyhydrodynamic. Moreover, we can say that a squirmer model isappropriate for expressing the motion of Paramecia, andhopefully for some other ciliates as well.

In most of the previous analytical studies on cell–cellinteractions (Guell et al., 1988; Ramia et al., 1993; Nasseriand Phan-Thien, 1997; Jiang et al., 2002), two cells in closecontact were not discussed. [Lega and Passot (Lega andPassot, 2003) included an ad hoc interactive force actingbetween cells, which in practice is unlikely to exist.] Thepresent results show, however, that the near-field interactiondramatically changes the orientation of cells. Sinceorientation change affects the macroscopic properties of thesuspension, such as the diffusivity, the near-field interactionhas to be solved accurately. The present results also show thatthe near-field interaction can be accurately solvedhydrodynamically by the boundary element method andlubrication theory.

List of symbolsAm surface area of squirmer mAR avoiding reactione orientation vector of cellER escape reactionFrep non-hydrodynamic interparticle repulsive forceh separation vectorHI hydrodynamic interactionK Oseen tensorL lengthq single-layer potential (traction force on the inner

surface minus that on the outer surface)Um translational velocity of squirmer mus tangential surface velocityus,m squirming velocity of squirmer mxm centre of squirmer mVm rotational velocity of squirmer ma1, a2 dimensionless coefficientse minimum separation between two surfacesu angle from the orientation vector of the cell

Appendix AVelocity disturbance due to a force-free particle near a wall

boundary

In Appendix A, we show that the velocity disturbance dueto a force-free particle decays as r–2, even though a no-slip wallboundary exists near the particle.

For an unbounded fluid, the Oseen tensor in Eqn·2 is given as:

where m is the viscosity, r=x–x9, x is the position vector of thefield point, and x9 is the point where the traction force isgenerated.

In order to account for a no-slip wall boundary, we exploit animage system for the Oseen tensor. The image system was derivedby Blake (Blake, 1971b), and a general outline of the method isincluded in standard texts (e.g. Kim and Karrila, 1992). If there isa wall boundary, the Oseen tensor needs to be rewritten as:

where R=x–X, and X is the position vector of the image pointthat is the mirror image of x9 about the wall boundary. Thedirection of subscript 3 is taken normal to the wall, and b is theminimum distance of x9 or X from the wall (for details, seefig.·1 in Blake, 1971b). Eqn·A2 indicates that the effect of the

⎛⎜⎝

⎛⎜⎝

⎞⎟⎠

rirjKij =

(�j3Ri + �i3Rj – 2�i3�j3R3)

2�i3�j3 – �ij + 3 (Rj–2�j3R3)

+r3

�ij

r

⎛⎜⎝

⎞⎟⎠

⎞⎟⎠

RiRj+

R3

2b+

R3

2b2

+R3

Ri

R2

�ij

R1

8��–

(A2)

⎡⎢⎣

,⎤⎥⎦

⎛⎜⎝

⎞⎟⎠

rirjKij = +

r3

�ij

r1

8��, (A1)


4462 T. Ishikawa and M. Hota

wall boundary is equivalent to inducing a point force, a dipoleand a source doublet at the image point.

When there are N particles and a wall boundary, the Stokesflow field around the particles can be given in an integral formas (similar to Eqn·2):

The right-hand side of Eqn·A3 can be expanded in momentsabout the centre of each particle and each image particle (seeDurlofsky et al., 1987):

where KK=K9–K, and F, L and S are, respectively, themonopole, the anti-symmetric dipole, and the symmetricdipole. Subscript a indicates real particles, and b indicatesimage particles. In the case of a force-free particle, Fa=Fb=0.Hence, the leading-order term in the right-hand side of Eqn·A4is r–2 or R–2. Therefore, the velocity disturbance due to a force-free particle decays as r–2, even though a no-slip wall boundaryexists near the particle.

Appendix BA boundary condition of constant swimming power

In Appendix B, we show that the difference between twoboundary conditions, constant surface velocity and constantswimming power, in the trajectories of two cells is small. Asimilar discussion has been published (Ishikawa et al., 2006).

Throughout this paper, the surface squirming velocity wasassumed not to change during the interactions. Real micro-organisms, however, may well change their swimming motionin the presence of a nearby micro-organism. Of course, we didnot model a cell’s biological response to other micro-organisms, but we can apply a different primitive boundarycondition for the squirmers to see if the effect is significant. Inthis boundary condition, the surface squirming velocity isdefined as lus. Here, us is the squirming velocity for a solitarysquirmer and l is a scalar factor that is chosen to realizeconstant swimming power, equal to the rate of viscous energy

⎛⎜⎝1 + �2

+

j

⎞⎟⎠

KijF� +

KijF� +

a2

6

⎛⎜⎝1 + �2

⎞⎟⎠

a2

6

⎛⎜⎝1 + �2

⎞⎟⎠

a2

10

⎛⎜⎝1 + �2

⎞⎟⎠

a2

10

ui(x) – �ui(x)� =

�lkjS(�kKil–�lKik)L� +

�lkjS(�kKil–�lKik)L� +

G(�kKij+�jKik)S� + ...

(A4)

�=1

N

1

8��

�=1

N

1

8��

⎡⎢⎣

⎡⎢⎣

⎤⎥⎦

j

jk

,G(�kKij+�jKik)S� + ...⎤⎥⎦

jk

j j

Am

⌠⎮⌡

m=1

K. q(x)dAm .u(x) = –

N

(A3)

dissipation throughout an interaction. The swimming powerconsumed by a squirmer is defined as:

where t is the traction force on the surface.We checked the relative translational–rotational velocities

under the constant-swimming-power condition. However, theeffect of changing the boundary condition was very small. Thiscan be explained as follows. The surface velocity isproportional to l, and the lubrication force is proportional tolog(e–1) and l, where e is the gap distance between two surfaces[for a detailed discussion about the lubrication force, seeIshikawa et al. (Ishikawa et al., 2006)]. The power is theproduct of these two quantities in the near-field, thusproportional to log(e–1) and l2. In order to maintain constantpower, l should therefore vary as:

Because log(e–1) is a very weak singularity, l changes veryslowly. Thus, the difference between the two boundaryconditions in the trajectories of two cells is small.

There may be some other boundary conditions for aswimming cell. Short et al., for instance, assumed constantsurface stress instead of constant surface velocity (Short et al.,2006). However, the effect of this boundary condition shouldalso be small, because the lubrication force between two cellsis again not sufficiently strong, even if e becomes the lengthscale of molecules, as mentioned above.

Appendix CEffect of a wall boundary on the swimming velocity of a

squirmer

In Appendix C, we show that the effect of a wall boundaryon the translational velocity of a squirmer parallel to the wallis small if the distance between the two surfaces e is larger than10–4 or 10–5.

We derived the first-order solution for the lubrication flowbetween a spherical squirmer and a sphere with arbitraryradius, in which a flat plate corresponds to infinitely largeradius (Ishikawa et al., 2006). Although the squirmer modelused in this study is spheroidal, we exploit the lubricationtheory for a spherical squirmer in the following discussion forsimplicity. The lubrication force due to the translationalmotion parallel to the wall is proportional to du log(e–1) to theleading order, where du is the velocity difference between thetwo surfaces in the lubrication region (cf. Ishikawa et al.,2006). du can be rewritten as du=U+us|lub, where U is thetranslational velocity of the squirmer parallel to the wall andus|lub is the squirming velocity in the lubrication region. Whenthe squirmer touches the wall, i.e. er0, du log(e–1) divergesto infinity unless du=0; thus, the squirmer swims with a

(B2). � 1log(�–1)

A

⌠⎮⌡

� . usdA , (B1)



velocity of –us|lub. When the surface squirming velocity isgiven by Eqn·1 and the orientation vector of the squirmer isparallel to the wall, the swimming velocity of the squirmerattached to the wall is about 1.5. Therefore, the wall boundaryincreases the swimming velocity of the squirmer by up to 50%when e=0.

Next, we show how du decays with e. The lubrication forceis generated by the relative motion between two surfaces inthe lubrication region. This force has to be canceled by theviscous drag force acting outside the lubrication region,because the particle is force-free. Since the viscous drag forceis not sensitive to e, we can assume that the lubrication forceis also unaffected by e to the leading order, i.e. dulog(e–1)=constant. Thus, du decays as 1/log(e–1). Asmentioned in Appendix B, log(e–1) is a very weak singularity[–log(10–4)=9.2 and –log(10–5)=11.5, for instance]. We canconclude that the effect of a wall boundary on thetranslational velocity of a squirmer parallel to the wall issmall if the distance between the two surfaces e is larger than10–4 or 10–5. In the present experiment, e is about 0.4, whichis much larger than 10–4.

The authors are grateful for helpful discussions with Prof.T. J. Pedley of DAMTP, University of Cambridge. Theauthors also appreciate Prof. Akitoshi Itoh of Tokyo DenkiUniversity for his kind help in dealing with the micro-organisms.

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interaction of two swimming paramecia · paramecium caudatumwas used in this study, because the...

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